Evaluation of
T and V after convergence.

At first we fix a convergence criterion e.g. 1.e-3. The
main goal is to mesure the temperature and velocity at a location which
caracterizes the regime. We choose the center of the box because this location
shows the greatest variations for V or T between under-critical and over-critical
regimes.However the aspect of residuals curves seems to be regular
in each regime :

Typical shape of residuals curves for under-critical
regime

Typical shape of residuals curves for over-critical regime

We noticed for under-critical regime the no convergence
of the computation even after extended run.Hence we decide to take into acount the mean value of
T and V on the set of iterations for which the residuals are levelled.

1) Evaluation of V.

Here we use the value (V) of
the vertical component of the velecity at the box center point to caracterize
the critical Rayleigh number. The evolution of V vs Ra number is drawn
below. We use the tangent at the inflexion point of the experimental curve
to extrapolate the curve to V = 0. This gives an estimated value of the
critical Ra.

We can plot the Velocity vs Ra number :

The evaluation of the critical Rayleigh number Rac is
made graphically using the inflexion point of the curve. The discrete set
of points does not permit us to determine the slope at the inflexion point.
Then we use the two lines defined by the neighbouring segments around
the discrete inflexion point.

We find 1572 < Rac < 1578 . This low
value compared to the admitted value of 1708 is due to the poor mesh refinement
whose effect has been describe in a previous report (Cayrol & Perchat):

2) Profile of T .

We defined a dimensionless temperature to be able
to compare measurement of T :

Tadim=(T-Tcold)/(Thot-Tcold)

The Rac is defined when Tadim=0.5
at the box center because it correspond to the value for linear temperature
gradient.We notice the flatness ot temperature profile when dT
tends to critical value.

We find Rac =1565 for the 05x10 mesh.

3) Conclusion.

The first remark is that we find different
Rac for the two methods. But the difference is less than 1%
between the two values.

Determination of
Rac
with V.We can see that the method is not
very accurate.The method tends to under-estimate
the Rac because the extrapolation method is not sufficiently
accurate.Determination
of Rac with Tadim.This method seems to be more reliable.
We seek the value of Tadim wich may tend to 0.5 for critical and under-critical
Ra.Nevertheless this yields to a slightly
lower value of Rac than using V.